In a measure theory course, we are given the following:
Let $X$ be a non-empty set and let $R$ be a collection of subsets of $X$. We say that $R$ is a $\pi$-system if it is closed under intersections, i.e. if it has the property: If $A \in R$ and $B \in R$ then $A \cap B \in R$.
This definition makes sense, but I'm not curious more generally about the meaning of "closed under intersection". Here I understand it to mean exactly what we have after the "i.e.", but should it mean more? We can have the empty intersection, which gives the whole of $X$, as explained here:
Which breaks what I understand to be the definition of a $\pi$-system. I'm not sure whether that makes the terminology "closed under intersections" incorrect, though. Does it? In general, if we say that, I'd have thought we should really mean closed under the empty intersection too, but is that something I should be concerned about?
There is a difference between arbitrary intersections (take a set, which could be empty or infinite, and intersect all the elements, usually denoted with a $\bigcap$ in front of the set) and pairwise intersections (take two elements and intersect them, usually denoted by the regular $\cap$ between the two sets).
My guess is that they're using the pairwise one (that's what their "i.e." is saying anyways), without stating it explicitly.